In a large bowl, cream the softened butter and granulated sugar until pale and light, then mix in the egg, vanilla, and milk.
0.625 cup unsalted butter, 0.667 cup granulated sugar, 1 large egg, 1 teaspoon vanilla extract, 2 tablespoons milk
Roll & Chill
On parchment paper, roll the dough into an 8×10-inch rectangle, spread the filling ingredients evenly, then roll tightly into logs.
2 tablespoons unsalted butter, 0.25 cup brown sugar
Wrap the logs in plastic and chill for at least 30 minutes to firm up.
Bake & Ice
Slice the chilled logs into ½-inch rounds, place on a parchment-lined baking sheet, and bake for 10–12 minutes until golden at the edges.
Whisk together the powdered sugar and milk until smooth, then drizzle the icing over cooled cookies.
2 teaspoons ground cinnamon
Notes
Testing Note #2: Chilling the rolled logs for at least 30 minutes keeps the swirls neat and stops filling seepage.Testing Note #3: Storing a slice of bread in the cookie container helps maintain a tender crumb overnight.